Chapter 13 - Davidson Physics

CHAPTER 14. FRACTALS AND KINETIC GROWTH MODELS 554a simple geometrical object (a circle or square) centered at the walker such that none of the clusteris within the object. The walker moves in one step to a site on the boundary of the object. Fora circle the walker can move to any location with equal probability on the circumference. Forthe square we need the probability of moving to various locations on the boundary. To find thelargest object that does not contain a part of the DLA cluster, consider coarse grained lattices.For example, each 2 × 2 group of sites on the original lattice corresponds to one site on the coarserlattice; each 2 × 2 group of sites on the coarse lattice corresponds to a site on an even coarserlattice, etc. If a site is occupied, then any coarse grained site containing this site also is occupied.a. Because we have considered DLA clusters on a square lattice, we use squares centered at thewalker. We first find the probability p(∆x, ∆y, s) that a walker centered on a square of lengthl = 2s + 1, will be displaced by the (∆x, ∆y). This probability can be computed by simulatinga random walk starting at the origin and ending at a boundary site of the square. Repeat thissimulation for many walkers, and then for various values of s. The fraction of walkers thatreach the position (∆x, ∆y) is p(∆x, ∆y, s). Determine p(∆x, ∆y, s) for s = 1 to 16. Store yourresults in a file.b. We next determine the arrays such that for a given value of s and a uniform random numberr, we can quickly find (∆x, ∆y). One way to do so is to create four arrays. The first array liststhe probability determined from part (a) such that the values for s = 1 are listed first. Call thisarray p. For example, p[1] = p(−1, −1, 1), p(2) = p(1)+p(−1, 0, 1), p[3] = p[2]+p(−1, 1, 1),etc. The array start tells us where to start in the array p for each value of s. The arraysdx(i) and dy(i) give the values of ∆x and ∆y corresponding to p[i]. To see how these arraysare used, consider a walker located at (x, y), centered on a square of linear dimension 2s + 1.Generate a random number r and find i = start(s). If r < p[i], then the walker moves to(x + dx(i), y + dy(i)). If not, increment i by unity and check again. Repeat until r ≤ p[i].Write a program to create these four arrays and store them in a file.c. Write a method to determine the maximum value of the parameter s such that a square of size2s + 1 centered at the position of the walker does not contain any part of the DLA cluster. Usecoarse grained lattices to do this determination more efficiently. Modify class DLA to incorporatethis method and the arrays defined part (b). How much faster is your modified program thanthe original class DLA for clusters of 500 and 5000 particles?d. What is the largest cluster you can grow on your computer in a reasonable time? Does thecluster show any evidence for anisotropy? For example, does the cluster tend to extend furtheralong the axes or along any other direction?Project 14.18. Cluster-cluster aggregationIn DLA all the particles that stick to a cluster are the same size (the growth occurs by the additionof one particle at a time), and the cluster that is formed is motionless. In the following, weconsider a cluster-cluster aggregation (CCA) model in which the clusters do a random walk asthey aggregate.Suppose we begin with a dilute collection of N particles. Each of these particles is initially acluster of unit mass and does a random walk until two particles become nearest neighbors. Theythen stick together to form a cluster of two particles. This new cluster now moves as a single